Directly coupled series of autoionising resonances

When two series of resonances are coupled directly, the K-matrix contains off-diagonal terms, and a mixed product containing poles of both series results. For the coupled system  

The resulting curve of v against energy, plotted modulo 1, exhibits a break near the energy E = Eb - C2Tb/2, where its slope is infinite, and re enters the graph on the opposite face. 

The expressions just given do not contain asymmetries due to the background continuum, which further increase the complexity of the problem. Their inclusion yields  

From equation (8.109), we deduce (modulo 1) that provided 1, 2 = 0  

7 Note that the poles in the numerator of (8.109) which occur when either Si + E2 — 00 or E1E2 — 00 are cancelled by corresponding poles in the denominator as noted above for equation (8.98) so that all the branches of the two-dimensional plot are described by equation (8.110). 

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